Question

Evaluate the integral: \[ \int \frac{x^2 (x \sec^2 x + \tan x)}{(x \tan x + 1)^2} dx \]

• A. \( -\frac{x^2}{x \tan x + 1} \)
• B. \( 2 \log_e |x \sin x + \cos x| + C \)
• C. \( -\frac{x^2}{x \tan x + 1} + 2 \log_e |x \sin x + \cos x| + C \)
• D. \( -\frac{x^2}{x \tan x + 1} - 2 \log_e |x \sin x + \cos x| + C \)

Hint:

Use integration by parts along with trigonometric identities to simplify complex integrals.